Douglas-Kroll-Hess method

Bioinorganic systems often contain heavy elements that need to be treated with an explicit relativistic method. It is now possible to carry out an explicit relativistic electronic structure calculation on the fly (152). The scalar-relativistic Douglas - Kroll - Hess method was implemented by us recently in the BOMD simulation framework (152). To use the relativistic densities in a non-relativistic gradient calculations turned out to be a valid approximation of relativistic gradients. An excellent agreement between optimized structures and geometries obtained from numerical gradients was observed with an error smaller than 0.02 pm. 

Apart from primary structural and energetic data, which can be extracted directly from four-component calculations, molecular properties, which connect measured and calculated quantities, are sought and obtained from response theory. In a pilot study, Visscher et al. (1997) used the four-component random-phase approximation for the calculation of frequency-dependent dipole polarizabilities for water, tin tetrahydride and the mercury atom. They demonstrated that for the mercury atom the frequency-dependent polarizability (in contrast with the static polarizability) cannot be well described by methods which treat relativistic effects as a perturbation. Thus, the varia-tionally stable one-component Douglas-Kroll-Hess method (Hess 1986) works better than perturbation theory, but differences to the four-component approach appear close to spin-forbidden transitions, where spin-orbit coupling, which the four-component approach implicitly takes care of, becomes important. Obviously, the random-phase approximation suffers from the lack of higher-order electron correlation. 

Mayer, M. (1999) A parallel density functional method implementation of the two-component Douglas-Kroll-Hess method and applications to relativistic effects in heavy-element chemistry. PhD thesis, Technical University of Munich. 

One-component calculations or two-component calculations including also spin-orbit coupling effects provide a firm basis for the calculations of higher-order relativistic corrections by means of perturbation theory. Several quasi-relativistic approximations have been proposed. The most successful approaches are the Douglas-Kroll-Hess method (DKH) [1-7], the relativistic direct perturbation theory (DPT) [8-24], the zeroth-order regular approximation (ZORA) [25-48], and the normalized elimination of small components methods (NESC) [49-53]. Related quasi-relativistic schemes based on the elimination of the small components (RESC) and other similar nonsingular quasi-relativistic Hamiltonians have also been proposed [54-61]. 

The lOTC method presented in this review is obviously related to earlier attempts to reduce the four-component Dirac formalism to computationally much simpler two-component schemes. Among the different to some extent competitive methods, the priority should be given to the Douglas-Kroll-Hess method [13,53,54]. It was Bernd Hess and his work which was our inspiration to search for the better solutions to the two-component methodology. 

T. Yoshizawa, M. Hada. Relativistic quantum-chemical calculations of mag-netizabilities of noble gas atoms using the Douglas-Kroll-Hess method. Chem. Phys. Lett, 458 (2008) 223-226. 

This is the equivalent of the second-order Douglas-KroU operator, but it only involves operators that have been defined in the free-particle Foldy-Wouthuysen transformation. As for the Douglas-Kroll transformed Hamiltonian, spin separation may be achieved with the use of the Dirac relation to define a spin-fi ee relativistic Hamiltonian, and an approximation in which the transformation of the two-electron integrals is neglected, as in the Douglas-Kroll-Hess method, may also be defined. Implementation of this approximation can be carried out in the same way as for the Douglas-Kroll approximation both approximations involve the evaluation of kinematic factors, which may be done by matrix methods. 

The transformed Hamiltonians that we have derived allow us to calculate intrinsic molecular properties, such as geometries and harmonic frequencies. We would like to be able to calculate response properties as well, with wave functions derived from the transformed Hamiltonian. If we used a method such as the Douglas-Kroll-Hess method, it would be tempting to simply evaluate the property using the nonrelativistic property operators and the transformed wave function. As we saw in section 15.3, the property operators can have a relativistic correction, and for properties sensitive to the environment close to the nuclei where the relativistic effects are strong, these corrections are likely to be significant. To ensure that we do not omit important effects, we must derive a transformed property operator, starting from the Dirac form of the property operator. 

Spin-free relativistic effects are readily incorporated into the ab initio model potential approximation by using a one-component spin-free relativistic method for the atom, such as the Cowan-Griffin method" or the Douglas-Kroll-Hess method. 

In the case of the Douglas-Kroll-Hess method, no special treatment of relativistic effects is required apart from the use of the one-electron Douglas-Kroll-Hess operator for the valence electrons. The direct and exchange potentials for the core electrons are treated in exactly the same way as in the nonrelativistic case but using the atomic Douglas-Kroll-Hess orbitals. However, only the unmodified part of the nuclear attraction is partitioned into a core and a valence part the core part is included with the core direct potential just as in the nonrelativistic case. 

The lowest-order effect of relativity on energetics of atoms and molecules—and hence usually the largest—is the spin-free relativistic effect (also called scalar relativity), which is dominated by the one-electron relativistic effect. For light atoms, this effect is relatively easily evaluated with the mass-velocity and Darwin operators of the Pauli Hamiltonian, or by direct perturbation theory. For heavier atoms, the Douglas-Kroll-Hess method or the NESC le method provide descriptions of the spin-independent relativistic effect that are satisfactory for all but the highest accuracy. 

Malkin E, Malkin I, Malkina OL, Malkin VG and Kanpp M 2006 Scalar relativistic calculations of hyperfine coupling tensors nsing the douglas-kroll-hess method with a finite-size nucleus model. Phys. Chem. Chem. Phys. 8, 4079-4085. 

We extend the method over all three rows of TMs. No systematic study is available for the heavier atoms, where relativistic effects are more prominent. Here, we use the Douglas-Kroll-Hess (DKH) Hamiltonian [14,15] to account for scalar relativistic effects. No systematic study of spin-orbit coupling has been performed but we show in a few examples how it will affect the results. A new basis set is used in these studies, which has been devised to be used with the DKH Hamiltonian. 

Molecules are more difficult to treat accurately than atoms, because of the reduced symmetry. An additional complication arises in relativistic calculations the Dirac-Fock-(-Breit) orbitals will in general be complex. One way to circumvent this difficulty is by the Douglas-Kroll-Hess transformation [57], which yields a one-component function with computational effort essentially equal to that of a nonrelativistic calculation. Spin-orbit interaction may then be added as a perturbation, implementation to AuH and Au2 has been reported [58]. Progress has also been made in the four-component formulation [59], and the MOLFDIR package [60] has been extended to include the CC method. Application to SnH4 has been described [61] here we present a recent calculation of several states of CdH and its ions [62], with one-, two-, and four-component methods. 

In a related study Ilias, Furdik and Urban have calculated FCu, FAg and FAu using the CCSD(T) method and considering relativistic effects by the nopair one-component Douglas-Kroll-Hess approximation. These are stable diatomic molecules in the S ground state with the bonding primarily arising from a s orbital formed by the 2p valence orbital of F and the ns valence orbital of the metal. 

The example of neon, where relativistic contributions account for as much as a0.5% of 711, shows that relativistic effects can turn out to be larger for high-order NLO properties and need to be included if aiming at high accuracy. Some efforts to implement linear and nonlinear response functions for two- and four-component methods and to account for relativity in response calculations using relativistic direct perturbation theory or the Douglas-Kroll-Hess Hamiltonian have started recently [131, 205, 206]. But presently, only few numerical investigations are available and it is unclear when it will become important to include relativistic effects for the frequency dispersion. 

Another group of methods successfully used for calculations of the electronic structures of the heaviest element molecules are effective core potentials (ECP) (see the Chapters of M. Dolg and Y.-S. Lee in these issues). The relativistic ECPs (RECP) were applied to calculations of the electronic structures of halides and oxyhalides of Rf and Sg and of some simple compounds (mostly hydrides and fluorides) of elements 113 through 118 [126-131]. Using energy-adjusted pseudo-potentials (PP) [132] electronic structures and properties, and the influence of relativistic effects were studied for a number of compounds of elements at the end of the 6d series (elements 111 and 112), as well as at the beginning of the 7p series (elements 113 and 114) (see Refs. 26 and 133 for reviews and references therein). Some other methods, like the Douglas-Kroll-Hess (DKH) [134], were also used for calculations of small heaviest-element species (e.g. IIIH [95]). 

Quantum chemistry with the Douglas-Kroll-Hess approach to relativistic density functional theory Efficient methods for molecules and materials... 

We review the Douglas-Kroll-Hess (DKH) approach to relativistic density functional calculations for molecular systems, also in comparison with other two-component approaches and four-component relativistic quantum chemistry methods. The scalar relativistic variant of the DKH method of solving the Dirac-Kohn-Sham problem is an efficient procedure for treating compounds of heavy elements including such complex systems as transition metal clusters, adsorption complexes, and solvated actinide compounds. This method allows routine ad-electron density functional calculations on heavy-element compounds and provides a reliable alternative to the popular approximate strategy based on relativistic effective core potentials. We discuss recent method development aimed at an efficient treatment of spin-orbit interaction in the DKH approach as well as calculations of g tensors. Comparison with results of four-component methods for small molecules reveals that, for many application problems, a two-component treatment of spin-orbit interaction can be competitive with these more precise procedures. 

A self-consistent scalar-relativistic (SR) version of the LCGTO-DF method has also been developed recently." "The SR variant employs a unitary second-order Douglas-Kroll-Hess (DKH) "" transformation for decoupling large and small components of the full four-component spinor solutions to the Dirac-Kohn-Sham equation. The approximate DKH transformation, very appropriate and efficient for molecular calculations, has been implemented this variant utilizes nuclear potential-based projectors and leaves the electron-electron interaction untransformed. 

Since it is actually impossible to sum the infinite series, this summation is terminated at particular numbers of unitary transformations 2 in DK2, 3 in DK3 and so forth. This method is also called the Douglas-Kroll-Hess transformation, because it was revised by Hess and coworkers (Jansen and Hess 1989). 

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